arXiv subjects

Thibaut Mastrolia

Publications and source records attributed to Thibaut Mastrolia.

At least 19 recordsLinked to original sources

arXiv2026

Learning Optimal Liquidation with Closing Auctions

We study liquidation when continuous trading is followed by a closing auction. The trader first sells through a limit-order book, then submits signed auction schedules to adjust the remaining inventory. A projected clearing price signal and intermediate auction feedback connect the two phases. We compare deep Q-network (DQN) policies with projected deep deterministic policy gradient (DDPG), twin delayed deep deterministic policy gradient (TD3) and soft actor-critic (SAC) policies, using synthetic rough Heston prices and historical midprice paths within a simulated market. Policies are selected and evaluated by inventory-penalized implementation shortfall, separately from a weighted training objective. They achieve lower inventory-penalized shortfall than the stylized Avellaneda-Stoikov (AS) and time-weighted average price (TWAP) references, and matched synthetic comparisons show that auction access is useful for all four learners. We furthermore find that dense auction credit improves learning; the clearing forecast is informative, but its incremental decision value is learner-dependent.

q-fin.TR↗

arXiv2026

Delegation or competition: optimal liquidation across dark and lit pools with information privilege

We study the optimal liquidation problem in both dark and lit pools for an investor delegating the execution of a large position to a broker, in a continuous-time setting with market impact and execution uncertainty. First, we design an optimal compensation scheme between the investor and a broker who owns dark pools, may internalize the client's order flow, and enjoys an information privilege over the rest of the market. We explicitly characterize the broker's optimal internalization, hedging, and routing strategies across the lit venue and the dark pools using BSDE theory. Second, we study a purely competitive model in which the investor liquidates directly on the open market against a crowd of strategic high-frequency traders, leading to a major-minor mean-field game for which we provide explicit solutions to the associated HJB Fokker-Planck system. Finally, we illustrate our results through numerical experiments, comparing the welfare and the price impact of the delegation regime with those of direct execution, and quantifying the economic value of broker intermediation.

q-fin.MF↗

arXiv2026

When Should a Human Take Back Control? Optimal Delegation under Turbulent AI Risk

Deploying AI systems requires deciding when to delegate tasks and when humans should intervene to monitor and mitigate risk induced by AI operations. These decisions are challenging when failures cluster: a hallucination or harmful output can trigger further errors, creating periods of elevated risk. We introduce a continuous-time framework for learning adaptive human oversight under such turbulent AI risk. Existing oversight and delegation formulations condition on history but do not model incident clustering, or its suppression by supervision effort, jointly with the delegation decision and this study fixes this gap. The self-exciting dynamics capture how risk events increase the likelihood of subsequent events, making their timing and history central to decision-making. We formulate a stochastic control problem combining human actions, monitoring effort, and switching between human-AI-assisted operation and full AI delegation, balancing operational rewards against oversight costs, and cascading AI-failures and induced uncertainty. Human participation is an endogenous component of risk management: the policy determines both when oversight is needed and how much effort to allocate. We study a relaxed switching formulation and propose Hawkes-PPO, a policy-gradient method that uses a bank of exponential filters of observed incident times. In a synthetic environment it attains a higher risk-adjusted objective than either fixed regime and approaches an approximate full-information oracle. We illustrate our results with numerical simulations by examining how cascade risks influence intervention and delegation, connecting reinforcement learning with adaptive human oversight of AI systems. In particular, we illustrate the benefit of our switching strategy and Hawkes-PPO algorithm to monitor the project efficiently along time, reducing turbulent risks occurrences and costs.

cs.LG↗

arXiv2026

Continuous-Time Reinforcement Learning for Controlled Hawkes Jump-Diffusions

We study stochastic control of multivariate Hawkes-driven stochastic differential equations with machine learning algorithms in a non-Markovian setting. Due to the path dependence of the memory of the Hawkes intensity, this problem does not fall within classical stochastic control theory outside particular Markovian kernels. We first develop a finite-dimensional Markovianization procedure and algorithm to approximate multivariate Hawkes processes with mixtures of exponential kernels. We prove the convergence of the Markovianized approximation of the Hawkes process, its intensity, and the value of the problem to the original non-Markovian processes and the value of the primal problem. We then formulate continuous-time deterministic policy gradient learning on the Markovianized approximation of the problem, called Hawkes-CT DDPG. We propose a model-free algorithm to solve the non-Markovian Hawkes-driven optimization by observing only the event times of the process, the realization of the solution to the SDE, and a chosen set of decay filters, while the Hawkes kernel coefficients remain unknown. We compare our continuous time reinforcement learning Hawkes-CT DDPG method with discrete time reinforcement learning techniques under three different types of kernels: simple exponential, Erlang, and power-law kernels.

cs.LG↗

arXiv2026

Signature Methods for Optimal Market Making

We propose a signature-based method to solve the optimal market-making problem under a mean-variance criterion. By exploiting signature linearization techniques, we reduce the market-making problem to a pseudo-linear optimization over the expected signature of an augmented market path, and we develop a signature algorithm named Sig-REINFORCE to learn the optimal bid and ask quotes. We test our method in two scenarios, in which market-order arrivals follow either a Poisson or a self-exciting Hawkes process, and we benchmark it against a Proximal Policy Optimization (PPO) baseline.

math.OC↗

arXiv2026

Optimal Impulse Control for Cyber Risk Management

We explore an optimal impulse control problem wherein an electronic device owner strategically calibrates protection levels against cyber attacks. Utilizing epidemiological compartment models, we qualitatively characterize the dynamics of cyber attacks within the network. We determine the optimal protective measures against effective hacking by formulating and solving a stochastic control problem with optimal switching. We demonstrate that the value function for the cluster owner constitutes a viscosity solution to a system of coupled variational inequalities associated with a fully coupled reflected backward stochastic differential equation (BSDE). Furthermore, we devise a comprehensive algorithm alongside a verification procedure to ascertain the optimal timing for network protection across various cyber attack scenarios. Our findings are illustrated through numerical approximations employing deep Galerkin methods for partial differential equations (PDEs). We visualize the optimal protection strategies in the context of two distinct attack scenarios: (1) a constant cyber attack, (2) an exogenous cyber attack strategy modeled with a Poisson process.

math.OC↗

arXiv2026

Optimal Rebate Design: Incentives, Competition and Efficiency in Auction Markets

This study explores the design of an efficient rebate policy in auction markets, focusing on a continuous-time setting with competition among market participants. In this model, a stock exchange collects transaction fees from auction investors executing block trades to buy or sell a risky asset, then redistributes these fees as rebates to competing market makers submitting limit orders. Market makers influence both the price at which the asset trades and their arrival intensity in the auction. We frame this problem as a principal-multi-agent problem and provide necessary and sufficient conditions to characterize the Nash equilibrium among market makers. The exchange's optimization problem is formulated as a high-dimensional Hamilton-Jacobi-Bellman equation with Poisson jump processes, which is solved using a verification result. To numerically compute the optimal rebate and transaction fee policies, we apply the Deep BSDE method. Our results show that optimal transaction fees and rebate structures improve market efficiency by narrowing the spread between the auction clearing price and the asset's fundamental value, while ensuring a minimal gain for both market makers indexed on the price of the asset on a coexisting limit order book.

q-fin.TR↗

arXiv2026

Delegated portfolio management with random default

We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. This uncertainty introduces significant complexities in analyzing the problem, requiring distinct mathematical approaches for two cases: when the random default time falls within the initial time frame [0,T] and when it extends beyond this period. We develop a theoretical framework to model the stochastic dynamics of the investment process, incorporating the random default time. We then analyze the portfolio manager's investment decisions and compensation mechanisms for both scenarios. In the first case, where the default time could be unbounded, we apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem. In the second case, where the default time is within the interval [0,T], the problem becomes more intricate due to the degeneracy of the BSDE's driver. For both scenarios, we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equations in both situations. We develop a deep-learning algorithm to solve the problem in high-dimension with no access to the optimizer of the Hamiltonian function.

q-fin.MF↗

arXiv2026

2BSDE with uncertain horizon and application to stochastic control in erratic environments

We investigate the existence and uniqueness of non-Markovian second-order backward stochastic differential equations with an uncertain terminal horizon and establish comparison principles under the assumption that the driver is Lipschitz continuous. The terminal time is both random and exogenous, and it may not be adapted to the Brownian filtration, leading to a singular jump in the 2BSDE decomposition. We also provide a connection between this new class of 2BSDE and a fully nonlinear PDE in a Markovian setting. Our theoretical results are applied to non-Markovian stochastic control problems in two settings: (1) when an agent seeks to maximize utility from a payoff received at an uncertain terminal time by controlling both the drift and volatility of a diffusion process; and (2) when the agent contends with volatility uncertainty stemming from an external source, referred to as Nature, and optimizes the drift in a worst-case scenario for the ambiguous volatility. We term this class of problems erratic stochastic control, reflecting the dual uncertainty in both model parameters and the timing of the terminal horizon.

math.PR↗

arXiv2026

Deep ZakaiJ: Structured Filtering for Jump-Diffusion Time Series Forecasting

Time series driven by unobserved latent states frequently exhibit abrupt jump discontinuities whose timing and magnitude cannot be predicted from observed history alone. Classical jump-diffusion models offer a principled mathematical framework but assume rigid parametric forms, while recent neural jump models operate on fully observed trajectories without inferring the hidden states that govern the dynamics. We propose \textit{Deep ZakaiJ}, a latent-state model for partially observed jump-diffusion systems that embeds the Zakai nonlinear filtering equation into a neural encoder--decoder architecture. The encoder recursively updates a belief over the latent state via Strang splitting into three interpretable substeps: prior propagation, diffusion innovation, and jump innovation, yielding a differentiable, first-order-accurate approximation of the exact filtering evolution. The decoder is a structured jump-diffusion model explicitly conditioned on the filtered belief, preserving the separation between continuous dynamics and discontinuous shocks. On synthetic, financial, and oceanographic datasets, \textit{Deep ZakaiJ} improves distributional forecasts while remaining competitive in point accuracy, achieving calibrated predictive intervals and recovering interpretable latent structure in synthetic and qualitative case studies.

cs.LG↗

arXiv2026

Agency Problems and Adversarial Bilevel Optimization under Uncertainty and Cyber Threats

We study an agency problem between a leader (the principal) seeking to design an optimal incentive scheme to a follower (the agent) to increase the value of a risky project subjected to accidents and volatility uncertainty. The agency problem is formulated as a max-min bilevel stochastic control problem with accidents and ambiguity. We show that the problem of the follower is reduced to solve a second order BSDE with jumps, reducing the problem of the leader to solve an integro-partial Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation. By extending stochastic Perron's method to our setting, we obtain viscosity sub- and supersolution envelopes for the Principal's integro-HJBI equation. Under an additional comparison principle in a suitable polynomial growth class, these envelopes coincide and the Principal's value is identified with the unique viscosity solution. The holding company seeks to design an optimal incentive scheme to mitigate these losses. In response, the subsidiary selects an optimal cybersecurity investment strategy, modeled through a stochastic epidemiological SIR (Susceptible-Infected-Recovered) framework. The cyber threat landscape is captured through an L-hop risk framework with two primary sources of risk, internal risk propagation via the contagion parameters in the SIR model, and external cyberattacks from a malicious external hacker. The uncertainty and adversarial nature of the hacking lead to consider a robust stochastic control approach that allows for increased volatility and ambiguity induced by cyber incidents. We illustrate our results with numerical simulations showing how the contracting mechanism enhances the quality of a cluster under cyber threats.

math.OC↗

arXiv2026

Approximation of Singular-Stopping Control Driven by Hawkes Processes via Rescaled MDPs

We investigate a singular-optimal stopping stochastic control problem driven by self-exciting dynamics governed by a Hawkes process. In the continuous-time setting, we show that the optimization problem reduces to solving a variational partial differential equation with gradient constraints. We then introduce its discrete-time counterpart, modeled as a Markov Decision Process. We prove that, under an appropriate rescaling procedure, the value function of the discrete-time problem converges to its continuous-time equivalent, implying that the discrete-time optimizers are asymptotically optimal for the continuous-time problem. Finally, we apply these results to an Ornstein-Uhlenbeck stochastic differential equation driven by a Hawkes process with singular control, motivated by optimal power plant investment under cyber threat and we illustrate the theoretical findings through numerical simulations.

math.OC↗

arXiv2024

Clearing time randomization and transaction fees for auction market design

Flaws of a continuous limit order book mechanism raise the question of whether a continuous trading session and a periodic auction session would bring better efficiency. This paper wants to go further in designing a periodic auction when both a continuous market and a periodic auction market are available to traders. In a periodic auction, we discover that a strategic trader could take advantage of the accumulated information available along the auction duration by arriving at the latest moment before the auction closes, increasing the price impact on the market. Such price impact moves the clearing price away from the efficient price and may disturb the efficiency of a periodic auction market. We thus propose and quantify the effect of two remedies to mitigate these flaws: randomizing the auction's closing time and optimally designing a transaction fees policy for both the strategic traders and other market participants. Our results show that these policies encourage a strategic trader to send their orders earlier to enhance the efficiency of the auction market, illustrated by data extracted from Alphabet and Apple stocks.

q-fin.TR↗

arXiv2023

Equilibria and incentives for illiquid auction markets

We study a toy two-player game for periodic double auction markets to generate liquidity. The game has imperfect information, which allows us to link market spreads with signal strength. We characterize Nash equilibria in cases with or without incentives from the exchange. This enables us to derive new insights about price formation and incentives design. We show in particular that without any incentives, the market is inefficient and does not lead to any trade between market participants. We however prove that quadratic fees indexed on each players half spread leads to a transaction and we propose a quantitative value for the optimal fees that the exchange has to propose in this model to generate liquidity.

q-fin.TR↗

arXiv2022

Agency problem and mean field system of agents with moral hazard, synergistic effects and accidents

We investigate the existence of an optimal policy to monitor a mean field systems of agents managing a risky project under moral hazard with accidents modeled by Lévy processes magnified by the law of the project. We provide a general method to find both a mean field equilibrium for the agents and the optimal compensation policy under general, sufficient and necessary assumptions on all the parameters. We formalize the problem as a bilevel optimization with the probabilistic version of a mean field games which can be reduced to a controlled McKean-Vlasov SDE with jumps. We apply our results to an optimal energy demand-response problem with a crowd of consumers subjected to power cut/shortage when the variability of the energy consumption is too high under endogenous or exogenous strains. In this example, we get explicit solution to the mean field game and to the McKean-Vlasov equation with jumps

math.OC↗

arXiv2022

Incentives, lockdown, and testing: from Thucydides's analysis to the COVID-19 pandemic

In this work, we provide a general mathematical formalism to study the optimal control of an epidemic, such as the COVID-19 pandemic, via incentives to lockdown and testing. In particular, we model the interplay between the government and the population as a principal-agent problem with moral hazard, à la Cvitanić, Possamaï, and Touzi [27], while an epidemic is spreading according to dynamics given by compartmental stochastic SIS or SIR models, as proposed respectively by Gray, Greenhalgh, Hu, Mao, and Pan [45] and Tornatore, Buccellato, and Vetro [88]. More precisely, to limit the spread of a virus, the population can decrease the transmission rate of the disease by reducing interactions between individuals. However, this effort, which cannot be perfectly monitored by the government, comes at social and monetary cost for the population. To mitigate this cost, and thus encourage the lockdown of the population, the government can put in place an incentive policy, in the form of a tax or subsidy. In addition, the government may also implement a testing policy in order to know more precisely the spread of the epidemic within the country, and to isolate infected individuals. In terms of technical results, we demonstrate the optimal form of the tax, indexed on the proportion of infected individuals, as well as the optimal effort of the population, namely the transmission rate chosen in response to this tax. The government's optimisation problem then boils down to solving an Hamilton-Jacobi-Bellman equation. Numerical results confirm that if a tax policy is implemented, the population is encouraged to significantly reduce its interactions. If the government also adjusts its testing policy, less effort is required on the population side, individuals can interact almost as usual, and the epidemic is largely contained by the targeted isolation of positively-tested individuals.

q-bio.PE↗

arXiv2021

On Z-mean reflected BSDEs

In this paper we provide conditions for the existence of supersolutions to BSDEs with mean-reflections on the $Z$ component. We show that, contrary to BSDEs with mean-reflections on the $Y$ component, we cannot expect a supersolution with a deterministic increasing process $K$. Nonetheless, we give conditions for the existence of a supersolution for a stochastic component $K$ and under various constraints. We formalize some previous arguments on the time-inconsistency of such problems, proving that a minimal supersolution is necessarily a solution in our framework.

math.PR↗

arXiv2020

AHEAD : Ad-Hoc Electronic Auction Design

We introduce a new matching design for financial transactions in an electronic market. In this mechanism, called ad-hoc electronic auction design (AHEAD), market participants can trade between themselves at a fixed price and trigger an auction when they are no longer satisfied with this fixed price. In this context, we prove that a Nash equilibrium is obtained between market participants. Furthermore, we are able to assess quantitatively the relevance of ad-hoc auctions and to compare them with periodic auctions and continuous limit order books. We show that from the investors' viewpoint, the microstructure of the asset is usually significantly improved when using AHEAD.

q-fin.TR↗